If magic is a dominant gene, then in theory squibs should never appear in pure-blood wizard families under normal circumstances. This means that when they do, it's because of a mutation ... or a parent having an affair with a muggle (who was probably coming back from a costume party or something). Same goes for muggle-born kids who can use magic, like Harry's friend Hermione: They're the product of illicit wizard-fucking, or they're mutants. Or both. Wizard-fuck-mutants, if you will.
This would explain her preference for other freaks of nature.
But if the gene is recessive, the game changes a lot -- it would mean that two magical parents would always have magical children, no matter what their background may be. All squibs would have to be mutants, and many generations of ordinary muggles could carry a recessive magical gene that might eventually pop up and make a muggle-born wizard-baby. This includes Harry's mom, meaning that Harry himself could have easily been a normal, boring English kid for whom the closest thing to a magical adventure was huffing glue.
The Harry Potter Lexicon
Harry may have gotten screwed when it comes to eyesight, but he was lucky otherwise.